Quaternion for transforming one frame to other?

Question

I am new to quaternions and learning how they can replace rotation matrices.

I know that we can use rotation matrices to describe a transformation from one frame to other. Where one may be a rotated frame and one fixed. Once we know that transformation matrix, a vector described in one frame can be represented in the other frame using that matrix.

How does this work with quaternions? All I can find out is that how to rotate a vector with them.

I am really looking for a quaternion that can describe the transformation from one frame to other. Just like we have for a rotation matrix.

Any help is appreciated.

Thanks

Answer 1

Well (in theory) being able to rotate vectors gives you all you want already. You just align the $x$-axes with a quaternion $q_1$, then align the $y$-axes with a quaternion $q_2$ which rotates around the aligned $x$-axes, and then $q_2q_1$ aligns the two frames.